Table of Contents

Diagnostic Tests. A Preview of Calculus. 1. FUNCTIONS AND MODELS. Four Ways to Represent a Function. Mathematical Models: A Catalog of Essential Functions. New Functions from Old Functions. Graphing Calculators and Computers. Exponential Functions. Inverse Functions and Logarithms. Review. Principles of Problem Solving. 2. LIMITS AND DERIVATIVES. The Tangent and Velocity Problems. The Limit of a Function. Calculating Limits Using the Limit Laws. The Precise Definition of a Limit. Continuity. Limits at Infinity; Horizontal Asymptotes. Derivatives and Rates of Change. Writing Project: Early Methods for Finding Tangents. The Derivative as a Function. Review. Problems Plus. 3. DIFFERENTIATION RULES. Derivatives of Polynomials and Exponential Functions. Applied Project: Building a Better Roller Coaster. The Product and Quotient Rules. Derivatives of Trigonometric Functions. The Chain Rule. Applied Project: Where Should a Pilot Start Descent? Implicit Differentiation. Derivatives of Logarithmic Functions. Rates of Change in the Natural and Social Sciences. Exponential Growth and Decay. Related Rates. Linear Approximations and Differentials. Laboratory Project: Taylor Polynomials. Hyperbolic Functions. Review. Problems Plus. 4. APPLICATIONS OF DIFFERENTIATION. Maximum and Minimum Values. Applied Project: The Calculus of Rainbows. The Mean Value Theorem. How Derivatives Affect the Shape of a Graph. Indeterminate Forms and L'Hospital's Rule. Writing Project: The Origins of l'Hospital's Rule. Summary of Curve Sketching. Graphing with Calculus and Calculators. Optimization Problems. Applied Project: The Shape of a Can. Newton's Method. Antiderivatives. Review. Problems Plus. 5. INTEGRALS. Areas and Distances. The Definite Integral. Discovery Project: Area Functions. The Fundamental Theorem of Calculus. Indefinite Integrals and the Net Change Theorem. Writing Project: Newton, Leibniz, and the Invention of Calculus. The Substitution Rule. Review. Problems Plus. 6. APPLICATIONS OF INTEGRATION. Areas between Curves. Volume. Volumes by Cylindrical Shells. Work. Average Value of a Function. Applied Project: Where to Sit at the Movies. Review. APPENDIXES. A. Numbers, Inequalities, and Absolute Values. B. Coordinate Geometry and Lines. C. Graphs of Second-Degree Equations. D. Trigonometry. E. Sigma Notation. F. Proofs of Theorems. G. The Logarithm Defined as an Integral. H. Complex Numbers. I. Answers to Odd-Numbered Exercises.